Conservation laws for the classical Toda field theories

TL;DR

This paper investigates conservation laws in classical Toda field theories, revealing an extensive class of models with infinite conserved charges, and highlights unique conservation laws in specific models like $A_m$.

Contribution

It provides explicit calculations of conservation laws in Toda models, identifies models with infinite conserved charges, and uncovers novel conservation laws not linked to known exponents or energy-momentum tensors.

Findings

Infinite conservation laws exist in a broad class of generalized Toda models.

Only $A_m$ and $A_m^{(1)}$ models admit spin-3 conservation laws.

Unique conservation laws in $A_m$ models for $m extgreater=4$ with different origins.

Abstract

We have performed some explicit calculations of the conservation laws for classical (affine) Toda field theories, and some generalizations of these models. We show that there is a huge class of generalized models which have an infinite set of conservation laws, with their integrated charges being in involution. Amongst these models we find that only the $A_m$ and $A_m^{(1)}$ ($m\ge 2$) Toda field theories admit such conservation laws for spin-3. We report on our explicit calculations of spin-4 and spin-5 conservation laws in the (affine) Toda models. Our perhaps most interesting finding is that there exist conservation laws in the $A_m$ models ($m\ge4)$ which have a different origin than the exponents of the corresponding affine theory or the energy-momentum tensor of a conformal theory.